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Keywords

Kirchhoff index; resistance distance.

Abstract

Let 𝑇 be a rooted tree, 𝐺 a connected graph, $x,y\in V(G)$ be fixed and $G_i$’s be $|V(T)|$ disjoint copies of 𝐺 with $x_i$ and $y_i$ denoting the corresponding copies of 𝑥 and 𝑦 in $G_i$, respectively. We define the 𝑇-repetition of 𝐺 to be the graph obtained by joining $y_i$ to $x_j$ for each $i\in V(T)$ and each child 𝑗 of 𝑖. In this paper, we compute the Kirchhoff index of the 𝑇-repetition of 𝐺 in terms of parameters of 𝑇 and 𝐺. Also we study how $Kf(G)$ behaves under some graph operations such as joining vertices or subdividing edges.